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Shimura (Crelle 221, 1966) considers the elliptic curve $E:y^2+y=x^3-x^2$ (although he doesn't use this equation) of conductor $11$ whose associated modular form is $$ q\prod_{k=1}^{+\infty}(1-q^k)^2(1-q^{11k})^2=\sum_{n=1}^{+\infty}c_nq^n $$ where $q=e^{2i\pi\tau}$ and $\tau$ is in the upper half of $\bf C$. For a prime $l$, he denotes by $K_l$ the extension of $\bf Q$ obtained by adjoining the $l$-torsion points of $E$ and shows that if $l\in[7,97]$, then ${\rm Gal}(K_l|{\bf Q})$ is isomorphic to ${\rm GL}_2({\bf F}_l)$.

Question. Is ${\rm Gal}(K_l|{\bf Q})$ now known to be isomorphic to ${\rm GL}_2({\bf F}_l)$ even for $l>97$ ?

Even if the faithful representation ${\rm Gal}(K_l|{\bf Q})\rightarrow{\rm GL}_2({\bf F}_l)$ fails to be surjective for a few $l>97$, does the recent proof of Serre's modularity conjecture not imply the

Statement. For every prime $l>5$ and every prime $p\neq11,l$, the characteristic polynomial of ${\rm Frob}_p$ (thought of as an element of ${\rm GL}_2({\bf F}_l)$) is $\equiv X^2-c_pX+p \pmod l$ ?

Shimura shows this only for $l\in[7,97]$.

Addendum. (2010/07/24) Looking at Shimura's paper beyond the first page shows that he actually proves (Section 3) that the characteristic polynomial for the action of ${\rm Frob}_p$ on the $l$-adic Tate module $T_l(E)$ is

$X^2-a_pX+p\in{\bf Z}_l[X]$

for all primes $l$ and $p\neq11, l$ and (Section 6) that $a_p=c_p$ for all $p\neq11$. And yes, he does use the Eichler-Shimura relation.

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1 Answer 1

up vote 5 down vote accepted

The first question is answered in Serre's [Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15:4 (1972) 259--331], on page 304, section 5.2, exactly for this curve. In general this paper give a good way to determine for which $\ell$ the mod-$\ell$ representation is not surjective. Sage can do that efficiently for a given curve.

For every prime $p$ different from $\ell$ and $11$, the characteristic polynomial of $\rm{Frob}_p$ is indeed $T^2 - c_p T +p$ in $ \mathbb F_{\ell}[T]$. The isomorphism $\rm{Gal}(K _{\ell}/\mathbb Q) \to \rm{Aut}(E[\ell])=\rm{GL} _2(\mathbb F _{\ell})$ sends $\rm{Frob}_p$ to the Frobenius endomorphism $\phi:E[\ell] \to E[\ell]$ on $E/\mathbb{F}_p$. Your $c_p$ is the trace of $ \phi$ and $p$ is the determinant of it since the Eichler--Shimura relation shows that $c_p$ is the Fourier coefficent of the associated modular form. See this answer for why it is so.

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Many thanks for for the reference to Serre: I should have looked it up before asking the question. What I found strange was that Shimura was claiming the "Statement" only for $l\in[7,97]$, whereas it seems to follows for all $l$ from the fact that the representation $\rho_l:{\rm Gal}(K_l|{\bf Q})\to{\rm GL}_2({\bf Z}_l)$ is unramfied at every $p\neq11,l$, that for these $p$ the characteristic polynomial of ${\rm Frob}_p\in{\rm GL}_2({\bf Z}_l)$ is $T^2-a_pT+p$ where $a_p$ is defined by ${\rm Card}(E({\bf F}_l))=1-a_p+p$, and finally the fact that $a_p=c_p$. –  Chandan Singh Dalawat Jul 20 '10 at 14:27
    
His method seems to need the restriction $l\in[7,97]$ for the surjectivity ${\rm Gal}(K_l|{\bf Q})\to{\rm GL}_2({\bf F}_l)$, though. –  Chandan Singh Dalawat Jul 20 '10 at 14:27
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I think the best reference is Cor.1 on p.308 of Serre's paper you mention. It says that if $E$ is a semistable elliptic curve over ${\bf Q}$ and if $p$ is the smallest prime where $E$ has good reduction, then ${\rm Gal}(K_l|{\bf Q})\to{\rm GL}_2({\bf F}_l)$ is surjective for every prime $l>(\sqrt p+1)^2$. –  Chandan Singh Dalawat Jul 20 '10 at 14:49

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